A complete analysis of the laser beam deflection systems used in cantilever-based systems

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1 Ultrmicroscopy 17 (7) A complete nlysis of the lser em deflection systems used in cntilever-sed systems L.Y. Beulieu,, Michel Godin, Olivier Lroche c, Vincent Trd-Coss c, Peter Gru tter c Deprtment of Physics nd Physicl Ocenogrphy, Memoril University. St. John s, NL., Cnd A1B 3X7 Division of Biologicl Engineering, Medi Lortory, Msschusetts Institute of Technology, Ames Street, Cmridge, MA 139, USA c Physics Deprtment, McGill University, Montrel, QC., Cnd H3A T8 Received Ferury 6; received in revised form Novemer 6; ccepted Novemer 6 Astrct A working model hs een developed which cn e used to significntly increse the ccurcy of cntilever deflection mesurements using opticl em techniques (used in cntilever-sed sensors nd tomic force microscopes), while simultneously simplifying their use. By using elementry geometric optics nd stndrd vector nlysis it is possile, without ny fitted or djustle prmeters, to completely nd ccurtely descrie the reltionship etween the cntilever deflection nd the signl mesured y position sensitive photo-detector. By rrnging the geometry of the cntilever/opticl em, it is possile to tilor the detection system to mke it more sensitive t different stges of the cntilever deflection or to simply linerize the reltionship etween the cntilever deflection nd the mesured detector signl. Supporting mteril nd softwre hs een mde ville for downlod t eulieu_l/ppers/cntilever_nlysis.htm so tht the reder my tke full dvntge of the model presented herein with miniml effort. r 6 Elsevier B.V. All rights reserved. Keywords: Cntilever; Opticl em deflection; Cntilever sensor; AFM 1. Introduction Micro-cntilevers re smll V-shped or rectngulr cntilevers (typiclly mde of silicon nitride (SiN x ) or silicon (Si)) which re of the order of 4 mm long, 5 mm wide nd 1 mm thick. Although micro-cntilevers re typiclly used in tomic force microscopes (AFM) for surfce imging, they hve lso een employed s ultrsensitive sensors to detect vrious phenomen such s chnges in temperture [1], chnges in mss [] nd the detection of chemicl rections through chnges in surfce stress [3]. Properly mesuring the cntilever deflection is t the hert of cquiring precision mesurements when performing cntilever-sed sensing or tomic force spectroscopy experiments. The most common method to mesure the cntilever deflection is y using n opticl em deflection Corresponding uthor. Tel.: ; fx: E-mil ddress: eulieu@physics.mun.c (L.Y. Beulieu). system [4]. Although there hve een some discussions in the literture regrding the lser detection scheme [5 15], we hve not seen ddressed nywhere the suject of how to otined well-defined reltionship etween the ctul cntilever deflection nd the PSD signl. Nor hve we seen nywhere discussed the influence of different opticl em deflection geometries on the reltionship etween the cntilever deflection nd the PSD signl. This is the suject of this pper. In recent pper, we hve reported how, y properly designing cntilever-sed instrument, it is possile to linerize the reltionship etween the cntilever deflection nd the mesurement mde with photo-sensitive detector (PSD) [16]. In this pper, we descrie in full detil the mthemticl model used to properly chrcterize the cntilever/lser em deflection system so tht potentil users my dpt nd use the model with their instruments. We lso provide we link where the reder cn downlod softwre nd other supporting mteril relted to this pper /$ - see front mtter r 6 Elsevier B.V. All rights reserved. doi:1.116/j.ultrmic

2 L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) Theory Fig. 1 shows schemtic representtion tht descries the geometry of cntilever/lser em detection system. In this digrm, the cntilever surfce is in the x y plne nd is oriented in the positive x direction. The cntilever chip is fixed in spce nd does not move. An incident lser hits the cntilever t distnce D from the se of the cntilever chip. The incident lser is fixed t n ngle of inclintion y with respect to the x y plne nd t n zimuthl ngle f mesured from the positive x-xis. The lser reflects off the free end of the cntilever into position sensitive detector (PSD) held t n initil distnce L from the cntilever. The PSD is itself inclined t n ngle x lso with respect to the x y plne. In Fig. 1, the line leled N c, is the vector norml to the surfce of the cntilever nd is used to clculte the reflected lser em direction in ccordnce with the lw of reflection. The vector eqution for stright line in three dimensions is given y I ¼ I p þ ti, (1) where I p is ny point on the line, I is the unit direction vector nd t is ny sclr. In this cse I p is defined s the initil point where the incident lser hits the cntilever t I p ¼ðD; ; Þ. Given the geometry shown in Fig. 1, the direction vector I is defined s I ¼ðcosðyÞ cosðp fþ; cosðyþ sinðp fþ; sinðyþþ. () During cntilever deflections, the incident lser em reflects wy from the surfce norml in ccordnce to the lw of reflection. Therefore it is impertive tht the proper cntilever curvture e tken into considertion. In cntilever sensor experiments, it hs een reported tht the cntilever undergoes deflection s if it were sujected to n end-moment [17,18]. For this type of deflection the curvture of the cntilever is descried y the following eqution: zðxþ ¼gx. (3) Fig. 1. Schemtic representtion of lser reflecting from cntilever into PSD detector. In this eqution, g is constnt defined y the physicl prmeters of the cntilever multiplied y the pplied end moment force. In the following clcultions, the end deflection of the cntilever is controlled which in turn governs the vlue of g. Therefore, t ech clculted deflection, g is defined s g ¼ zðx mxþ x. (4) mx The vlue of x mx is determined y solving the integrl eqution, which defines the totl length CL of the cntilever. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z xmx CL ¼ 1 u t vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z xmx ¼ 1 þ zðx! u t mxþ x dx x mx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ zðx mxþ x mx þ 4zðx mxþ x mx lnðx mxþ 4zðx mx Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ x mx lnðzðx mxþþ x mx þ 4zðx mxþ Þ ð5þ 4zðx mx Þ Solving for x mx from Eq. (5) is done numericlly. During AFM force spectroscopy experiments, point lod is pplied to the end of the lever. Such force cuses cntilever to deflect with curvture s descried y Eq. (6) [19] zðxþ ¼ Fx ðx 3Þ, (6) 6EI where F is the pplied force, E is the Young s Modulus of elsticity, I is the re moment of inerti, nd is the loction of the pplied force. As efore, in order to perform the simultions, the cntilever curvture needs only to e descried using Eq. (7). zðxþ ¼gx ðx 3Þ, (7) where g is determined y the mximum deflection. zðx mx Þ g ¼ x 3 mx. (8) 3x mx The vlue of x mx is determined y solving the length eqution vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z xmx CL ¼ 1 þ qz! u t dx qx Z xmx rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ ðgð3x 6xÞÞ dx ð9þ the solution of which is too lengthy to show here. Solving this eqution numericlly gives the coordinte of the end of the lever (x mx, z end ). When the cntilever deflects, the intersection point W etween the incident lser nd the cntilever surfce must e clculted. This is ccomplished y solving the

3 44 vector eqution ARTICLE IN PRESS L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) 4 43 Tle 1 I ¼ I p þ ti ¼ðW x ; W y ; gw x ÞÞ ¼ W ðend moment deflectionþ, I ¼ I p þ ti ¼ðW x ; W y ; gw x ðw x 3ÞÞ ¼ W ðpoint lod deflectionþ. ð1þ At the contct point W etween the incident lser nd the cntilever, it is necessry to otin the vector norml N c to the cntilever surfce. The slope m NC of the vector N c is given y the negtive of the reciprocl of the slope of the cntilever t the point W. This cn e written s m NC ¼ 1 ðend moment deflectionþ, (11) gw x 1 m NC ¼ gð3w x 6W xþ ðpoint lod deflectionþ. (1) Defining the ngle ¼ rctnðm NC Þ, the vector N c my e written s N c ¼ðcosðÞ; ; sinðþþ. (13) The direction vector of the reflected em R is y definition, the reflection of I cross N c in the I N c plne. This is well-known formul [] given y Eq. (14). R ¼ I ði N c ÞN c. (14) As in the cse of the incident em, the vector line of the reflected em R is given y the vector eqution R ¼ W þ tr. (15) As one of the djustle prmeters, the PSD/cntilever seprtion is given y the vlue L. The vlue of L is used to define the initil contct point PSD p of the reflected lser on the PSD PSD p ¼ I p þ LR. (16) Fig.. Schemtic representtion of the vector geometry involved in chrcterizing the plne of the PSD. The vector PSD n is norml to the PSD plne while the vectors PSD 1 nd PSD re in the PSD plne. In this digrm, the vector R represents the reflected lser em, which is incident on the PSD t the point PSDp. The vector R sutends n ngle with respect to the PSD surfce norml. For f ¼ 1 For f ¼ 91 For f ¼ 181 PSD 1 ¼ðcosðxÞ; ; sinðxþþ, PSD ¼ð; 1; Þ PSD 1 ¼ð; cosðxþ; sinðxþþ, PSD ¼ð1; ; Þ PSD 1 ¼ð cosðxþ; ; sinðxþþ, PSD ¼ð; 1; Þ ð17þ ð18þ ð19þ The vector PSD p is determined t the eginning of the lgorithm nd remins fixed for the durtion of the clcultions. The ctive re of the PSD is descried s plne in spce. Descriing such plne requires point PSD p nd norml vector PSD n (see Fig. ). The norml vector PSD n is determined y two orthogonl vectors defined y the geometry in Fig. 1. Fig. shows schemtic representtion of the geometry used to descrie the PSD. In this digrm, the vector PSD n is norml to the PSD plne while the vectors PSD 1 nd PSD re in the PSD plne. Also, since the PSD plne is inclined with respect to the x y plne, the vector PSD n my not necessrily e prllel to the reflected lser em vector R. In ll clcultions, the vectors I, R, nd PSD n ll lie in the sme plne. The vectors descriing the PSD plne re shown in Tle 1 for three specific cses. Finlly, the vector norml to the PSD plne is given y the following vector product: PSD n ¼ PSD PSD 1. () Once the PSD plne hs een defined, it is then necessry to determine the intersection point rpsd etween the PSD plne nd the reflected lser R. This is done y using the following eqution []: rpsd ¼ W þ ðpsd p WÞPSD n R. (1) R PSD n The lgorithm descried ove is performed for rnge of cntilever deflections (i.e. from to some end deflection). Since the PSD plne is inclined in 3D spce, it is necessry to rotte it such s to trnsform it into D spce. This is shown schemticlly in Fig. 3. The generl rottion mtrix M used to rotte ny vector y n ngle out vector u is given y the following []: u x þ Cð1 u x Þ u 3 xu y ð1 CÞ u z S u x u z ð1 CÞþu y S M ¼ u x u y ð1 CÞþu z S u y þ Cð1 u y Þ u 6 yu z ð1 CÞ u x S 7 4 5, u x u z ð1 CÞ u y S u y u z ð1 CÞþu x S u z þ Cð1 u z Þ () where Ccos() ndssin(). Although ¼ p/ x, the vlue of u chnges depending on which system is eing studied. Tle shows the different vlues of u. Throughout this pper the linerity of the PSD mesurement versus cntilever deflection curves is discussed.

4 L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) Results Fig. 3. Schemtic representtion showing how the PSD plne in 3D is rotted y n ngle d ¼ p/ x into D plne. Tle System Vlue of f Vector u 1 f ¼ 1 u ¼ð; 1; Þ f ¼ 91 u ¼ð 1; ; Þ 3 f ¼ 181 u ¼ð; 1; Þ In order to compre the linerity of different dt we will quote the vlue of w given y the following: w ¼ X ðrpsdðz n ðx mx ÞÞ mz n ðx mx ÞÞ, (3) n where rpsdðz n ðx mx ÞÞ is the clculted PSD mesurement for specific cntilever deflection z n ðx mx Þ t the end point of the cntilever x mx nd m is the slope of the est fit line. 3. Equipment Mtching experiments performed to vlidte the ove model were conducted on mcro-sized cntilever mde in-house. The rectngulr-shped cntilever ws constructed from 3.17 mm thick luminum piece, 5.4 mm wide with n extended unsupported length of 41 mm. The cntilever ws held horizontlly pproximtely mm from solid luminum plte in which holes cple of ccepting dowel pins were mchined t precise loctions such s to force the cntilever to end with the desired curvture s required y different ending mechnisms. A stndrd undergrdute lortory HeNe lser ws used to monitor the deflection of the cntilever. In order to promote the reflection of the lser off the cntilever surfce, the free end of the lever ws mchined polished to mirror finish. The lrge size of this system mde the mesurement of ech prmeter D, L, CL, f, y, nd x possile with high precision. Fig. 4 shows experimentl (symols) nd clculted dt (solid lines) of the PSD displcement versus cntilever deflection for system 3 descried in Tle. The PSD signl is given in units of length which descries the displcement of the incident lser spot on the PSD. Most cntilever-sed instruments use PSD where mesured voltge is linerly proportionl to the displcement of the lser on the PSD. Hence the PSD displcement versus cntilever deflection dt s shown here is consistent with current technology. The two plots 4 () nd () show dt tken where the vlues of D, L, f nd y re fixed nd x vries from 1 to 91. As the dt shows, excellent greement is otined etween the experimentl nd clculted dt. It is importnt to stress tht the ove model completely nd ccurtely descries the mesured dt without the use of ny djustle or fitted prmeters. The vlues of D, L, CL, f, y, nd x uniquely defines the cntilever/lser em detection system. Fig. 4 shows how the concvity of the dt chnges s the vlue of x is chnged (similr results occur if D, L, f nd x re fixed nd y is vried). The PSD versus cntilever deflection curve goes from concve to convex s the PSD ngle is chnged from.1 to 9.1, respectively. The chnge in curvture cn esily e rtionlized when the geometry of the reflected lser nd the ngle of the PSD re tken into considertion. As the PSD/Cntilever deflection curve goes PSD Signl (mm) ξ=. ξ = 16.1 ξ =. ξ = ξ = 5.16 ξ = 67. ξ = Cntilever Deflection (mm) Fig. 4. Comprison etween experimentl (symols) nd clculted dt (solid lines) for the PSD displcement versus cntilever deflection of system 3 descried in Tle for different vlues of PSD ngle () x ¼ :1; 16:11; :1; 39:171 nd () x ¼ 5:161; 67:1; 9:1. The mcrocntilever ws forced to deflect y n end-moment. The fixed geometricl prmeters of the system were: D ¼ 38:5 :5 mm, L ¼ 63:7 :5mm, CL ¼ 41: :5 mm, f ¼ , y ¼

5 46 L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) 4 43 from concve to convex, it psses region where the reltionship is nerly liner. We hve found tht for ny geometry D, L, CL, f, y, nd ending curvture, it is possile to find PSD ngle x (or x fixed nd y is vried) such tht the linerity of the reltionship etween the PSD nd the cntilever deflection is mximized. As n exmple, the geometry: CL ¼ 375 mm, D ¼ 35 mm, L ¼ 3: cm, f ¼ 181 (system 3), nd y ¼ 71 is chrcteristic of typicl cntilever sensor setup used in our lortory. For this geometry, ssuming mximum cntilever deflection of 4 mm, it cn e found y lestsqures-fit tht the optimized PSD ngle is x ¼ 37:51. Fig. 5 shows comprison etween clculted dt (gry crosses) nd liner fit (solid line) showing the linerity of the PSD signl. Fig. 5 shows, on much-reduced scle, the difference etween the optimized liner PSD/cntilever curve nd the stright line fit. At most, difference of 4 nm corresponds, s shown in Fig. 5c to n error of.8% of the mximum PSD mesurement. In fct, clcultions hve shown tht regrdless of the vlues of D, L, f, nd y, it is lwys possile to find vlue of x tht mximizes the linerity of the reltionship etween the PSD signl nd the cntilever deflection. When ttempting to linerize the PSD versus cntilever deflection reltionship, it is importnt to note tht not only does the vlue of x depend on the prmeters D, L, CL, f, nd y, ut it lso depends on the mximum cntilever deflection (rnge). Fig. 6) shows how the optimized PSD ngle x vries with the mximum cntilever deflection. The reson for this dependence stems from the fct tht the reltionship etween the PSD displcement nd the cntilever deflection is not perfectly liner s shown in Fig. 5. The PSD displcement/cntilever deflection curve oscilltes, lthough minutely, out the est-fit line used to otin the PSD ngle x. Therefore, depending on the deflection rnge used in the nlysis, the vlue of x will differ. However the dt shows tht the vlue of x ecomes sttionry for smll cntilever deflections (o3 mm) which is the rnge where most cntilever-sed experiments occur. Fig. 6 revels tht w lso decreses s the cntilever deflection rnge decreses. This indictes tht the ccurcy of the fit increses when smller cntilever deflections re considered. However, even for lrge deflections, the linerity of the optimized geometry is still dequte s shown y the smll vlue of w. As the cntilever deflects, the position where the lser em hits the cntilever lso chnges. As expected for system 3 (Tle ), the x-position (see Fig. 1) of the intersection point of the incident lser em on the cntilever (the x-coordinte of the vector W) increses with cntilever deflection. Our model cn e used to completely descrie the position of the lser spot on the cntilever for ny deflection. Fig. 7 shows plot of the displcement of the intersection point etween the incident lser em nd the cntilever s mesured directly long the length of the lever versus cntilever deflection. The grph shows comprison etween mesured dt (lck PSD Displcement (mm) Difference (μm) Percent Error c Cntilever Defelction (μm) Fig. 5. () Clculted dt (gry crosses) nd liner fit (solid line) for n optimized setup with CL ¼ 375 mm, D ¼ 35 mm, L ¼ 3: cm, f ¼ 181, y ¼ 71 nd x ¼ 37:51. () Difference etween the clculted dt nd est line of fit. (c) Percent error of the linerity fit. PSD Angle ξ(deg.) χ Mximum Cntilever Deflection (μm) Fig. 6. () Clculted results showing how for the following geometry, CL ¼ 375 mm, D ¼ 35 mm, L ¼ 3: cm, f ¼ 181, y ¼ 71, the optimized ngle of the PSD chnges with the mximum cntilever deflection. () Clculted results showing how w increses when the mximum cntilever deflection increses. This result indictes tht the reltionship etween the PSD signl nd the cntilever deflection for n optimized geometry ecomes incresing liner when smll mximum cntilever deflections re considered.

6 L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) circles) s tken with the mcro-cntilever nd the results of the model (solid gry line). Clerly the model completely nd ccurtely descries the position of the lser spot on the cntilever. Agin it is importnt to emphsize tht no djustle prmeters re used nywhere in this nlysis. For the cse of lser reflecting off cntilever in typicl AFM, n end deflection of 1 mm would cuse the incident lser to undergo totl displcement of pproximtely 3 mm on the cntilever surfce. Although system 3 (Tle ) is the most commonly used geometry for cntilever-sed instruments, there re other geometries tht cn e used to monitor the cntilever deflection. Fig. 8 shows experimentl dt (symols) tken with the mcro-cntilever nd clculted curves from the model of the PSD signl versus cntilever deflection for the following geometries: f ¼ 1, y ¼ 69:91, L ¼ 63:7 mm, D ¼ 383:5 mm, CL ¼ 41 mm nd x ¼ 9:551; 39:751; nd 59:641. As in the previous cse the model perfectly descries the f ¼ 1 system. The min difference etween systems 1 nd 3 is in the reltionship etween the incident lser em I nd the norml vector N c. In the cse of system 3, the ngle etween I nd N c continuously increses while for system 1 the ngle continuously decreses during cntilever deflections. As consequence, system 1 imposes more physicl restrictions for it requires tht the PSD nd the lser focusers e in close proximity to ech other since the ngle of incidence (nd reflection) ecomes smller s the cntilever ends. A second consequence imposed y physicl restriction is tht it is more difficult to otin strongly convex reltionship etween the PSD signl nd the cntilever deflection then for system 3. However the opposite is true for producing concve reltionship. As in the cse for f ¼ 181, it is possile to linerize the reltionship etween the PSD displcement nd cntilever deflection for system 1 (f ¼ 1) y optimizing the PSD ngle x (or incident lser ngle y). Fig. 9 shows clculted dt for f ¼ 1, y ¼ 71, L ¼ 3 cm, D ¼ 35 mm, CL ¼ 35 mm nd n optimized PSD ngle of x ¼ 13:891. These vlues re typicl dimensions for ctul cntilever sensors. Fig. 9 shows the clculted PSD displcement (gry symols) versus cntilever deflection for the optimized geometry long with liner est-fit line (solid line). The difference etween these two curves is shown on reduced scle in Fig. 9 nd the reltive percent error in Fig. 9c. With w ¼ :4, these vlues re comprle to those found for the f ¼ 181 system nd represent n uncertinty well within experimentl noise. Also similr to the f ¼ 181 system, it is possile to model the motion of the lser spot on the cntilever s the cntilever is forced to deflect. For this cse however, s the cntilever deflects the lser spot moves up the length of the cntilever (closer to the chip). As in the previous cse, excellent greement ws otined etween experimentl nd clculted vlues. Both systems 1 nd 3 (Tle ) re very similr nd differ only y their geometry. Specificlly oth show similr ehvior nd ccurcy regrdless of the types of cntilever deflections used. However, system chrcterized y f ¼ 91 (see Fig. 1) is very different thn systems 1 nd 3 (f ¼ 1 nd 181, respectively) where the length of the cntilever, the incident nd reflected lser re ll restricted to the x z plne. For the cse of f ¼ 91, the incident lser is prllel to the y z plne. Becuse of this, the PSD signl is no longer stright line ut curve s shown schemticlly in Fig. 1. Notice tht for the f ¼ 91 system, the PSD must e rotted so tht the ctive region of the detector (long xis) is oriented in the PSD direction. In contrst, for the f ¼ 1 nd 181 systems, the PSD is oriented so tht the long xis of the PSD is in the PSD 1 direction. The ove model cn e used to simulte the trce mde on the PSD y the reflected lser s the cntilever is mde to deflect. Fig. 11 shows clculted dt of the long xis (PSD direction) versus the short xis Position on Cntilever (mm) PSD Displcement (mm) ξ = 9.55 ξ = ξ = Cntilever Deflection (mm) Fig. 7. Position of the lser spot on the cntilever versus cntilever deflection. Dots show experimentl dt tken with mcro-cntilever while the solid curve is clculted dt from the model discussed within the text. In this dt, the geometry of the mcro cntilever ws set to f ¼ 181, y ¼ 69:91, D ¼ 383:5 mm nd CL ¼ 41 mm Cntilever Deflection (mm) Fig. 8. Experimentl (symols) nd clculted PSD displcement (solid curves) versus cntilever deflection for the following geometries: f ¼ 1, y ¼ 69:91, L ¼ 63:7 mm, D ¼ 383:5 mm, CL ¼ 41 mm nd x ¼ 9:551; 39:751 nd

7 48 ARTICLE IN PRESS L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) PSD Displcement (mm) Difference (μm) Percent Error c Cntilever Deflection (μm) Fig. 9. () Clculted dt (gry crosses) nd liner fit (solid line) of PSD displcement versus cntilever deflection for the optimized geometry: f ¼ 1, y ¼ 71, L ¼ 3 cm, D ¼ 35 mm, CL ¼ 35 mm nd n optimized PSD ngle of x ¼ 13:881. () Difference nd (c) reltive percent error etween the clculted dt nd the liner fit shown in (). (-PSD 1 direction) of the PSD displcement (see Fig. 1) for the following geometries: f ¼ 91, y ¼ 71, L ¼ 3 cm, D ¼ 35 mm, CL ¼ 35 mm nd x ¼ 1:1; x ¼ 5:1; x ¼ 6:1; x ¼ 7:1ndx ¼ 8:541. For these clcultions the cntilever ws mde to deflect s if sujected to n end moment to mximum deflection of 4 mm. As shown, the curvture of the PSD trce increses with incresing vlues of x. The sme results re otined for clcultions performed with point lod type deflection. However for the cse of the point lod, the curvture t the free end of the cntilever where the lser is mde to reflect is much smller then for end moment-type deflections. As consequence the curving effects re slightly smller. The geometry descried y system hs to dte only een used in cntilever-sed sensor instruments [1]. The PSD used in most cntilever sensors re only sensitive to the displcement of the reflected lser long the length (long xis) of the PSD. Therefore, provided the reflected lser spot remins within the ctive re of the PSD (pprox. 1 (long) (wide) mm) the mesurement is unffected y the curvture of the trce mde y the reflected lser. Tking this under considertion it is possile to optimize the f ¼ 91 system y linerizing the long-xis component of the trce mde y the reflected lser versus the cntilever deflection. Fig. 11 shows the long-xis component of the PSD trce versus cntilever deflection of the sme dt shown in Fig. 11. Although ech curve is close to eing liner, the optimized PSD ngle Fig. 1. Schemtic representtion showing the trce mde on the PSD y the reflected lser for the f ¼ 91 system. In this geometry the PSD must e rotted so tht the ctive region of the PSD is long the PSD direction. PSD Displcement (mm) (PSD direction) ξ = 1. ξ = 5. ξ = 6. ξ = 7. ξ = Cntilever Deflection (μm) PSD Displcement (mm) (- PSD 1 direction) Cntilever Deflection (μm) Fig. 11. () Clculted results showing the trce of the incident lser em on the PSD during cntilever deflections for vrious PSD ngles x for the following geometries: f ¼ 91, y ¼ 71, L ¼ 3 cm, D ¼ 35 mm, CL ¼ 35 mm nd x ¼ 1:1; 5:1; 6:1; 7:1 nd () The long xis component (PSD direction) of the lser em trce on the PSD s function of cntilever deflection for the sme PSD ngles shown in pnel (). (c) Expnded view of the PSD versus cntilever deflection curves shown in pnel () with the ssocited PSD ngles x. is found to e with w ¼ 7: Fig. 11c shows the smll differences etween the five different curves shown in Fig. 11. It is only t lrge cntilever deflections such s 4 mm tht the difference etween the five curves is pprecile. In most cntilever-sed experiments the cntilever is generlly only mde to deflect few micrometers. For the cse of system with f ¼ 91, y ¼ 71, L ¼ 3 cm, D ¼ 35 mm, CL ¼ 35 mm with cntilever sujected to point lod t 3 mm, the optimized PSD ngle is x ¼ 81:41 with w ¼ 5: In order to further vlidte the ove model, we show in Fig. 1 comprison of the PSD displcement of oth the short- nd long-xis components versus cntilever deflection etween experimentl mesurements (symols) collected with mcro-cntilever nd clculted dt (solid curves) otined from the model. The geometry of the PSD Displcement (mm) c

8 L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) PSD Displcement (mm) ξ = 9.55 ξ = ξ = 69.9 Long xis direction Short xis direction moment, point-lod, nd/or uniform lod type deflections re induced y very different physicl phenomen. The ove model hs een incorported into n excel spredsheet nd visul sic progrm for end-moment nd point lod type deflections, respectively. We hve lso included visul sic progrm to solve for x mx in oth Eqs. (5) nd (9). This supporting mteril hs een mde ville for downlod off our wesite t: physics.mun.c/eulieu_l/ppers/cntilever_nlysis.htm. We welcome contct from users tht would like to implement the ove model into their instrument or s prt of their dt nlysis Conclusion Cntilever Deflection (mm) Fig. 1. Comprison etween experimentl mesurements (symols) collected with mcro-cntilever nd clculted dt (solid curves) otined from the model descried within the text. The geometry of the system ws s follows: f ¼ 91, y ¼ 69:91, L ¼ 63:7 mm, D ¼ 383:5 mm, CL ¼ 41 mm nd x ¼ 9:551; 49:111 nd system ws s follows: f ¼ 91, y ¼ 69:91, L ¼ 63:7 mm, D ¼ 383:5 mm, CL ¼ 41 mm nd x ¼ 9:551; 49:111 nd 69:91. As efore, the greement etween the experimentl dt nd our model is very good. In this cse however, there is some noise in the experimentl dt, which origintes from the inility to reduce torque effects on the mcro-cntilever when forcing deflections. In this pper, we hve stressed the fct tht for ny kind of cntilever deflection nd cntilever/lser em detection system geometry it is possile to djust the incident lser ngle or the PSD ngle such tht the reltionship etween the PSD displcement nd the cntilever deflection is nerly liner. It is sometimes desirle tht this reltionship not e liner ut curved. For exmple, Figs. 4 nd show how PSD held t n ngle x ¼ :1 produces concve curvture while PSD held t n ngle x ¼ 9:1 produces convex curvture. Clerly the concve curvture will provide more sensitive dt t lrger deflections while the convex curvture will provide more sensitive mesurements t smller cntilever deflections. This cn e very useful when ttempting to detect smll deflection signl within lrger mesurement rnge. Also due to system restrictions, it is not lwys possile to orient the incident lser ngle or PSD ngle in order to linerize the system. However s long s it is possile to otin the vlues of D, L, CL, f, y, nd x (for exmple, from mnufcture specifictions), the ove model cn e used to otined the exct reltionship etween the cntilever deflection nd PSD signl provided the proper cntilever curvture is tken into considertion. This lst condition is not very stringent ecuse end- We hve shown y using elementry geometric optics nd vector nlysis tht it is possile to completely chrcterize the cntilever/lser em deflection system so s to otin the exct reltionship etween the cntilever deflection nd the PSD mesurement. We hve lso shown tht y djusting either the incident lser ngle or the PSD ngle it is possile to tilor the reltionship etween the cntilever deflection nd PSD displcement so s to mke it more sensitive t different cntilever deflections or nerly liner to increse its ese of use. Elsewhere we hve shown how the ove model cn e incorported with new equipment design so s to otin self-clirting system [16]. We elieve tht the nlysis presented here nd in Ref. [16] should e strongly considered when designing the next genertion of cntilever-sed sensor instruments or tomic force microscopes. Acknowledgments The uthors would like to thnk NSERC, CFI, IRIF, Memoril University, Genome Queec, nd McGill University for funding through vrious progrms, grnts nd fellowships. References [1] Gimzewshi, et l., Chem. Phys. Lett. 17 (1994) 598. [] Dvis, et l., J. Vc. Sci. Technol. B 18 (1996) 61. [3] Berger, et l., Science 76 (1997) 1. [4] G. Meyer, N.M. Amer, Appl. Phys. Lett. 53 (4) (1988). [5] C.A.J. Putmn, B.G. De Grooth, N.F. Vn Hulst, J. Greve, J. Appl. Phys. 7 (1) (199). [6] C.A.J. Putmn, B.G. de Grooth, N.F. vn Hulst, J. Greve, Ultrmicroscopy (199) 4. [7] A. Grci-Vlenzuel, J. Villtoro, J. Appl. Phys. 84 (1) (1998). [8] G. Moulrd, G. Contoux, G. Grdet, G. Motyl, M. Couron, Surf. Cot. Technol. 97 (1997). [9] T. Miytni, M. Fujihir, J. Appl. Phys. 81 (11) (1997). [1] C. Kylner, L. Mttsson, Rev. Sci. Instrum. 68 (1) (1997). [11] Z. Hu, T. Seeley, S. Kossek, T. Thundt, Rev. Sci. Instrum. 75 () (4). [1] R. Riteri, H.-J. Butt, M. Grttrol, Electrochim. Act 46 (). [13] N.P. D Cost, J.H. Hoh, Rev. Sci. Instrum. 66 (1) (1995).

9 43 ARTICLE IN PRESS L.Y. Beulieu et l. / Ultrmicroscopy 17 (7) 4 43 [14] G. Moulrd, G. Contous, G. Motyl, G. Grdet, M. Couron, J. Vc. Sci. Technol. A 16 () (1998). [15] S. Fujisw, H. Ogiso, Rev. Sci. Instrum. 74 (1) (3). [16] L.Y. Beulieu, M. Godin, O. Lroche, V. Trd-Coss, P. Gru tter, Appl. Phys. Lett. 88 (6) [17] G.G. Stoney, Proc. R. Soc. Lond. Ser. A 8 (17) (199). [18] R.W. Hoffmn, Physics of Thin Films 3, Acdemic, New York, 1966, p. 11. [19] D. Srid, Scnning Force Microscopy: With Applictions to Electric, Mgnetic, nd Atomic Forces, Oxford University Press, Jnury [] G.E. Frin, D. Hnsford, The Geometry Toolox for Grphics nd Modeling, Ntick, Mss, [1] M. Godin, O. Lroche, V. Trd-Coss, L.Y. Beulieu, P. Gru tter, P.J. Willims, Rev. Sci. Instrum. 74 (3) 49.